A GROUP THEORETIC APPROACH TO THE EQUATIONS ...marsden/bib/1982/06-Ma1982/...Canad. Math. Bull. Vol....
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, \ , .' Canad. Math. Bull. Vol. 25 (2), 1982 A GROUP THEORETIC APPROACH TO THE EQUATIONS OF PLASMA PHYSICS JERROLD E. MARSDEN· ABSTRACT. This paper concerns the interaction between group theory and classical mechanics in general and with the application of this theory to plasma physics in particular. Mechanics has had a profound role in the historical development of mathematics. Leading original thinkers in pure mathematics such as Newton, Euler, Lagrange, Jacobi, Laplace, Cauchy, Gauss, Riemann. Poincare, Hilbert, Birkhoff, Smale and Arnold were also great original thinkers in various facets of mechanics. The interac- tion hetween mechanics and pure mathematics remains one of the most active and flourishing areas of current research. II is this flavor which I hope to convey in this lecture. Acknowledgements. Much of the work that I will descri'?e below was done in collaboration with Alan Weinstein (see Marsden and [1981]). A seminal paper of l\1orrison [1980] originally shown to us by Allan Kaufman was our starting point. Our ideas were developed in a series of lectures given by both of us to Kaufman's plasma physics-dynamics seminar early in 1981 at Berkeley. The participant's enthusiasm and encouragement were very impor- tant. I thank the officers of the Canadian Mathematical Society for the opportun- ity and honor to present these ideas in the Jeffrey-Williams lecture. I wish to dedicate this lecture to my undergraduate teachers from Toronto who had such a positive influence on my career, especially Professors Atkinson, Coxeter, Davis, Duff, Pillow, Rooney, Rund, Scherk, Scherk and Vanstone. It was through them that I first appreciated the fact that pure applied mathematics have a magnificant intersection within mathematics. § 1. Introduction. Commencing around 1750, Euler discovered and investi- gated the basic equations governing two of the most important mechanical systems, namely the rigid body and a perfect incompressible fluid. He would have been pleased with the fundamental paper of Arnold [1966] which showed Received by the editors September 20, 1981. * This paper is based on the JelJrey-Williams lecture given at the meeting of the Canadian Mathematical Society, Dalhousie University. Halifax, Nova Scotia, May 28, 1981 upon the invitation of the Research Committee of the CMS. Research partially supported by NSF grant MCS-78-06718, ARO contract DAAG-29-79-C-0086, and the Miller Institute for Basic Re- search. AMS (1980) Classification: 58D. 58E, 58F, 70H. 70W. 129
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Canad. Math. Bull. Vol. 25 (2), 1982
A GROUP THEORETIC APPROACH TO THEEQUATIONS OF PLASMA PHYSICS
JERROLD E. MARSDEN·
ABSTRACT. This paper concerns the interaction between grouptheory and classical mechanics in general and with the application ofthis theory to plasma physics in particular.
Mechanics has had a profound role in the historical developmentof mathematics. Leading original thinkers in pure mathematics suchas Newton, Euler, Lagrange, Jacobi, Laplace, Cauchy, Gauss,Riemann. Poincare, Hilbert, Birkhoff, Smale and Arnold were alsogreat original thinkers in various facets of mechanics. The interac-tion hetween mechanics and pure mathematics remains one of themost active and flourishing areas of current research. II is this flavorwhich I hope to convey in this lecture.
Acknowledgements. Much of the work that I will descri'?e below was donein collaboration with Alan Weinstein (see Marsden and Wei~stein [1981]). Aseminal paper of l\1orrison [1980] originally shown to us by Allan Kaufmanwas our starting point. Our ideas were developed in a series of lectures givenby both of us to Kaufman's plasma physics-dynamics seminar early in 1981 atBerkeley. The participant's enthusiasm and encouragement were very impor-tant.
I thank the officers of the Canadian Mathematical Society for the opportun-ity and honor to present these ideas in the Jeffrey-Williams lecture.
I wish to dedicate this lecture to my undergraduate teachers from Torontowho had such a positive influence on my career, especially Professors Atkinson,Coxeter, Davis, Duff, Pillow, Rooney, Rund, Scherk, Scherk and Vanstone.It was through them that I first appreciated the fact that pure an~ appliedmathematics have a magnificant intersection within mathematics.
§ 1. Introduction. Commencing around 1750, Euler discovered and investi-gated the basic equations governing two of the most important mechanicalsystems, namely the rigid body and a perfect incompressible fluid. He wouldhave been pleased with the fundamental paper of Arnold [1966] which showed
Received by the editors September 20, 1981.* This paper is based on the JelJrey-Williams lecture given at the meeting of the Canadian
Mathematical Society, Dalhousie University. Halifax, Nova Scotia, May 28, 1981 upon theinvitation of the Research Committee of the CMS. Research partially supported by NSF grantMCS-78-06718, ARO contract DAAG-29-79-C-0086, and the Miller Institute for Basic Re-search.
AMS (1980) Classification: 58D. 58E, 58F, 70H. 70W.
129
